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Evaluate the limit:
$$\mathop {\lim }\limits_{x \to \infty } x\left( {{{\left( {1 + {1 \over x}} \right)}^x} - e} \right)$$

My attempts didn't yield a result. I'd be glad for a guidance.

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Have you tried bringing $x$ to the dominator, as $\frac{1}{x}$? – user25004 Jun 3 '14 at 14:11
Did you repost this question? Or undelete it? – draks ... Jun 3 '14 at 14:36
@draks... I complained about the deletion to the moderators. – Ted Shifrin Jun 3 '14 at 14:38
up vote 13 down vote accepted

Ugh @ L'Hôpital's Rule :)

Let $f(h) = \begin{cases} (1+h)^{1/h}, & h\ne 0 \\ e, & h=0 \end{cases}$. Then the limit you want is $f'(0)$.

(To verify this, we note, substituting $h=1/x$, that $$\lim_{x\to\infty} x\left(\Big(1+\frac1x\Big)^x - e\right) = \lim_{h\to 0^+} \frac{(1+h)^{1/h}-e}h = \lim_{h\to 0^+} \frac{f(h)-f(0)}h.)$$

Using a Taylor polynomial, one checks that $\ln f(h) = \dfrac1h\ln(1+1/h) = 1-\frac12 h + \epsilon(h)$, where $\lim\limits_{h\to 0}\dfrac{\epsilon(h)}h=0$, so $$\frac{f'(0)}{f(0)} = (\ln f)'(0) = -\frac12.$$ Therefore, $f'(0) = -e/2$.

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How is $f^{\prime}(0)$ the same limit ? – Rene Schipperus Jun 3 '14 at 14:45
OK thanks, you wrote you initial $f$ wrong, but I thinks its a better solution than mine. – Rene Schipperus Jun 3 '14 at 14:54
Ohhhh... Oops. Typo. Thanks, @Rene. – Ted Shifrin Jun 3 '14 at 14:56
Thanks also for getting this question undeleted, its definitely a worthwhile addition to the site. – Rene Schipperus Jun 3 '14 at 15:00

There is a nice little trick to do this.

$$\lim\limits_{h\rightarrow 0+}{\frac{1}{h}\left((1+h)^{\frac{1}{h}}-e\right)}$$ $$(1+h)^{\frac{1}{h}}=\exp{\left(\frac{1}{h}\ln{(1+h)}\right)}=\exp\left({\frac{h-\frac{h^2}{2}+o(h^2)}{h}}\right)=e\left(1-\frac{h}{2}+o(h)\right)$$ Hence the limit is reduced to. $$\frac{1}{h}\left((1+h)^{\frac{1}{h}}-e\right)=e\left(-\frac{1}{2}+o(1)\right)\longrightarrow -\frac{e}{2}$$ as $h\rightarrow 0^+$.

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you should use exp in the appropriate places I believe, now you mix e as a number and e(x) as a function. – yo' Jun 3 '14 at 16:20
Yeah, this needs to be fixed I totally read the answer wrong. – Rene Schipperus Jun 3 '14 at 16:42

You just have to persist in the L'Hopital rule. Start with

$$\frac{(1+\frac{1}{x})^x-e}{\frac{1}{x}}$$ top and bottom go to zero, by L'Hopital we get

$$\frac{(\ln(1+\frac{1}{x})-\frac{1}{x+1})(1+\frac{1}{x})^x}{-\frac{1}{x^2}} =(\frac{x^2}{x+1}-x^2\ln(1+\frac{1}{x}))(1+\frac{1}{x})^x$$

Now $(1+\frac{1}{x})^x\to e$ so lets set that term aside. What remains can be written $$\frac{\frac{1}{x+1}-\ln(1+\frac{1}{x})}{\frac{1}{x^2}}$$ top and bottom go to zero so again with L'Hopital, $$\frac{-\frac{1}{(x+1)^2}+\frac{x}{x+1}\frac{1}{x^2}}{-\frac{2}{x^3}}=-\frac{1}{2}(\frac{x^2}{x+1}-\frac{x^3}{(x+1)^2})=-\frac{1}{2}\frac{x^2}{(x+1)^2}\rightarrow -\frac{1}{2}$$

So we get the limit $-\frac{e}{2}$

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For your amusement, what's $f''(0)$? Still want to use L'Hôpital's rule? :) – Ted Shifrin Jun 3 '14 at 14:59

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